Zeta function of representations of compact 𝑝-adic analytic groups

Jaikin-Zapirain, A.

doi:10.1090/S0894-0347-05-00501-1

Zeta function of representations of compact $p$-adic analytic groups
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J. Amer. Math. Soc. 19 (2006), 91-118 Request permission

Abstract:

Let $G$ be an FAb compact $p$-adic analytic group and suppose that $p>2$ or $p=2$ and $G$ is uniform. We prove that there are natural numbers $n_1, \ldots , n_k$ and functions $f_1(p^{-s}),\ldots , f_k(p^{-s})$ rational in $p^{-s}$ such that \[ \zeta ^G(s)=\sum _{\lambda \in \operatorname {Irr}(G)} \lambda (1) ^{-s}=\sum _{i=1}^kn_i^{-s}f_i(p^{-s}).\]

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